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A288791 Number of blocks of size >= nine in all set partitions of n. 2

%I

%S 1,11,122,1245,12325,121136,1195147,11915997,120572790,1241499241,

%T 13030331671,139549798524,1525923634907,17041290249637,

%U 194394900237176,2264977282222371,26951265841776186,327445918493429897,4060993235341162405,51396034231430455550

%N Number of blocks of size >= nine in all set partitions of n.

%H Alois P. Heinz, <a href="/A288791/b288791.txt">Table of n, a(n) for n = 9..575</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F a(n) = Bell(n+1) - Sum_{j=0..8} binomial(n,j) * Bell(n-j).

%F a(n) = Sum_{j=0..n-9} binomial(n,j) * Bell(j).

%F E.g.f.: (exp(x) - Sum_{k=0..8} x^k/k!) * exp(exp(x) - 1). - _Ilya Gutkovskiy_, Jun 26 2022

%p b:= proc(n) option remember; `if`(n=0, 1, add(

%p b(n-j)*binomial(n-1, j-1), j=1..n))

%p end:

%p g:= proc(n, k) option remember; `if`(n<k, 0,

%p g(n, k+1) +binomial(n, k)*b(n-k))

%p end:

%p a:= n-> g(n, 9):

%p seq(a(n), n=9..30);

%t Table[Sum[Binomial[n, j] BellB[j], {j, 0, n - 9}], {n, 9, 30}] (* _Indranil Ghosh_, Jul 06 2017 *)

%o (Python)

%o from sympy import bell, binomial

%o def a(n): return sum([binomial(n, j)*bell(j) for j in range(n - 8)])

%o print([a(n) for n in range(9, 31)]) # _Indranil Ghosh_, Jul 06 2017

%Y Column k=9 of A283424.

%Y Cf. A000110.

%K nonn

%O 9,2

%A _Alois P. Heinz_, Jun 15 2017

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Last modified August 10 04:27 EDT 2022. Contains 356029 sequences. (Running on oeis4.)